-- produce a nullhomotopy for a map f of chain complexes.
Whether f is null homotopic is not checked.
Here is part of an example provided by Luchezar Avramov. We construct a random module over a complete intersection, resolve it over the polynomial ring, and produce a null homotopy for the map that is multiplication by one of the defining equations for the complete intersection.
i1 : A = ZZ/101[x,y];
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i2 : M = cokernel random(A^3, A^{-2,-2})
o2 = cokernel | 2x2+40xy-22y2 9x2-20xy-31y2 |
| 46x2+25xy+33y2 35x2+16xy+44y2 |
| x2-39xy-46y2 -29x2-23xy-11y2 |
3
o2 : A-module, quotient of A
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i3 : R = cokernel matrix {{x^3,y^4}}
o3 = cokernel | x3 y4 |
1
o3 : A-module, quotient of A
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i4 : N = prune (M**R)
o4 = cokernel | 3x2+42xy-39y2 -35x2+27xy-46y2 x3 x2y-3xy2 8xy2-17y3 y4 0 0 |
| x2-3xy-20y2 -39xy+44y2 0 -38xy2-12y3 -21xy2+30y3 0 y4 0 |
| -17xy-15y2 x2+36xy+38y2 0 -41y3 xy2+20y3 0 0 y4 |
3
o4 : A-module, quotient of A
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i5 : C = resolution N
3 8 5
o5 = A <-- A <-- A <-- 0
0 1 2 3
o5 : ChainComplex
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i6 : d = C.dd
3 8
o6 = 0 : A <------------------------------------------------------------------------- A : 1
| 3x2+42xy-39y2 -35x2+27xy-46y2 x3 x2y-3xy2 8xy2-17y3 y4 0 0 |
| x2-3xy-20y2 -39xy+44y2 0 -38xy2-12y3 -21xy2+30y3 0 y4 0 |
| -17xy-15y2 x2+36xy+38y2 0 -41y3 xy2+20y3 0 0 y4 |
8 5
1 : A <------------------------------------------------------------------------- A : 2
{2} | 14xy2+20y3 -44xy2+26y3 -14y3 25y3 31y3 |
{2} | -31xy2+19y3 -50y3 31y3 6y3 -21y3 |
{3} | 46xy-19y2 49xy+3y2 -46y2 4y2 48y2 |
{3} | -46x2+21xy+5y2 -49x2-18xy+8y2 46xy-2y2 -4xy-31y2 -48xy-29y2 |
{3} | 31x2+41xy-26y2 20xy+40y2 -31xy+41y2 -6xy-37y2 21xy+5y2 |
{4} | 0 0 x-39y 16y 25y |
{4} | 0 0 31y x-36y 36y |
{4} | 0 0 33y 20y x-26y |
5
2 : A <----- 0 : 3
0
o6 : ChainComplexMap
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i7 : s = nullhomotopy (x^3 * id_C)
8 3
o7 = 1 : A <------------------------ A : 0
{2} | 0 x+3y 39y |
{2} | 0 17y x-36y |
{3} | 1 -3 35 |
{3} | 0 39 10 |
{3} | 0 -41 2 |
{4} | 0 0 0 |
{4} | 0 0 0 |
{4} | 0 0 0 |
5 8
2 : A <-------------------------------------------------------------------------- A : 1
{5} | -28 26 0 5y -13x-21y xy-32y2 23xy+17y2 -30xy+22y2 |
{5} | -43 50 0 -33x-7y 39x+6y 38y2 xy+44y2 21xy+7y2 |
{5} | 0 0 0 0 0 x2+39xy+14y2 -16xy+7y2 -25xy-39y2 |
{5} | 0 0 0 0 0 -31xy-26y2 x2+36xy-13y2 -36xy-43y2 |
{5} | 0 0 0 0 0 -33xy-10y2 -20xy-5y2 x2+26xy-y2 |
5
3 : 0 <----- A : 2
0
o7 : ChainComplexMap
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i8 : s*d + d*s
3 3
o8 = 0 : A <---------------- A : 0
| x3 0 0 |
| 0 x3 0 |
| 0 0 x3 |
8 8
1 : A <----------------------------------- A : 1
{2} | x3 0 0 0 0 0 0 0 |
{2} | 0 x3 0 0 0 0 0 0 |
{3} | 0 0 x3 0 0 0 0 0 |
{3} | 0 0 0 x3 0 0 0 0 |
{3} | 0 0 0 0 x3 0 0 0 |
{4} | 0 0 0 0 0 x3 0 0 |
{4} | 0 0 0 0 0 0 x3 0 |
{4} | 0 0 0 0 0 0 0 x3 |
5 5
2 : A <-------------------------- A : 2
{5} | x3 0 0 0 0 |
{5} | 0 x3 0 0 0 |
{5} | 0 0 x3 0 0 |
{5} | 0 0 0 x3 0 |
{5} | 0 0 0 0 x3 |
3 : 0 <----- 0 : 3
0
o8 : ChainComplexMap
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i9 : s^2
5 3
o9 = 2 : A <----- A : 0
0
8
3 : 0 <----- A : 1
0
o9 : ChainComplexMap
|